Weighted Residual Numerical Differentiation Algorithm Applied to Experimental Bending Moment Data

A weighted-residual approach for differentiating one-dimensional discrete data is presented and applied to an experimental program in which distributions of bending moment were measured along a model pile foundation in a centrifuge test. The weighted-residual approach is validated by first differentiating a sinusoidal bending moment distribution, and errors in first and second derivatives associated with various ratios of wavelength to sampling interval are computed. A bending moment distribution from a finite-element simulation of a pile foundation is differentiated using the weighted-residual technique, by fitting cubic splines, and by polynomial regression, and second derivatives are compared with the recorded subgrade reaction distributions. The influence of adding noise to the sampled bending moment distribution prior to differentiation is explored and is found to be most influential when sampling intervals are small. Bending moment data recorded during the centrifuge experiment are double differentiated and uncertainty in strain gauge calibration factors and position are incorporated using a Monte Carlo simulation to assess potential errors in the computed second derivatives.